Nonradiality of second fractional eigenfunctions of thin annuli

TL;DR

This paper investigates the properties of second fractional eigenfunctions in annuli-like domains, demonstrating nonradiality for large radii and analyzing eigenvalue maximization in specific geometric configurations.

Contribution

It establishes nonradiality of second eigenfunctions in large annuli and characterizes the maximization of second eigenvalues in domains with holes, extending understanding of fractional Laplacian eigenfunctions.

Findings

Second eigenfunctions are nonradial for large annuli.

The second eigenvalue is maximized at the center in certain domains.

The first part confirms the existence of nonradial eigenfunctions in large annuli.

Abstract

In the present paper, we study properties of the second Dirichlet eigenvalue of the fractional Laplacian of annuli-like domains and the corresponding eigenfunctions. In the first part, we consider an annulus with inner radius $R$ and outer radius $R+1$. We show that for $R$ sufficiently large any corresponding second eigenfunction of this annulus is nonradial. In the second part, we investigate the second eigenvalue in domains of the form $B_1(0)\setminus \overline{B_{\tau}(a)}$, where $a$ is in the unitary ball and $0<\tau<1-|a|$. We show that this value is maximized for $a=0$, if the set $B_1(0)\setminus \overline{B_{\tau}(0)}$ has no radial second eigenfunction. We emphasize that the first part of our paper implies that this assumption is indeed nonempty.